Science:Math Exam Resources/Courses/MATH100/December 2010/Question 01 (k)

MATH100 December 2010

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Question 01 (k)
If $\displaystyle f(0)=10$ and $f'(x)\geq 3$ for $0\leq x\leq 4$, what is the least $\displaystyle f(4)$ could possibly be?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Let's look at all the information we have. We know that we are looking at a function on an interval $\displaystyle [0,4]$ where we know something about its derivative and about the value of the function at 0. Do we have a formula or a theorem that can relate all these values?

Hint 2
Try the mean value theorem.

Hint 3
The mean value theorem states the following: If $\displaystyle f(x)$ is continuous on $\displaystyle [a,b]$ and differentiable on $\displaystyle (a,b)$, then there exists a point $\displaystyle c\in [a,b]$ such that $\displaystyle f'(c)={\frac {f(b)-f(a)}{b-a}}$.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Notice that $\displaystyle f(x)$ is differentiable on $\displaystyle [0,4]$ and hence it is continuous there. Thus, by the mean value theorem, there exists a point $\displaystyle c\in [0,4]$ such that $\displaystyle f'(c)={\frac {f(4)-f(0)}{4-0}}$. We know that on $\displaystyle [0,4]$ that $\displaystyle f'(x)\geq 3$. Since $\displaystyle c\in [0,4]$ and $\displaystyle f(0)=10$ , we have that $\displaystyle 3\leq f'(c)={\frac {f(4)-f(0)}{4-0}}={\frac {f(4)-10}{4}}$. Solving gives $\displaystyle 22\leq f(4)$ and hence the minimum $\displaystyle f(4)$ could be is 22.

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